week ending 13 JUNE 2014
PHYSICAL REVIEW LETTERS
PRL 112, 230601 (2014)
Accelerating search kinetics by following boundaries 1
T. Calandre,1 O. Bénichou,1 and R. Voituriez1,2 Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), CNRS / UPMC, 4 Place Jussieu, 75255 Paris Cedex 2 Laboratoire Jean Perrin (FRE 3231) CNRS /UPMC, 4 Place Jussieu, 75255 Paris Cedex (Received 4 March 2014; published 9 June 2014) We derive exact expressions of the mean first-passage time to a bulk target for a random searcher that performs boundary-mediated diffusion in a circular domain. Although nonintuitive for bulk targets, it is found that boundary excursions, if fast enough, can minimize the search time. A scaling analysis generalizes these findings to domains of arbitrary shapes and underlines their robustness. Overall, these results provide a generic mechanism of optimization of search kinetics in interfacial systems, which could have important implications in chemical physics. In the context of animal behavior sciences, it shows that following the boundaries of a domain can accelerate a search process, and therefore suggests that thigmotactism could be a kinetically efficient behavior. DOI: 10.1103/PhysRevLett.112.230601
PACS numbers: 05.40.Fb, 82.20.−w, 87.23.−n
In interfacial systems, such as porous media, or biopolymer [1–3], micellar and colloidal [4–6] systems, the transport of a generic tracer particle combines diffusion phases in the bulk and diffusion phases at interfaces due to transient binding events, which can now be observed at the single molecule scale [7,8]. Interestingly, similar intermittent transport patterns have also been observed at a larger scale in the context of animal movement, where numerous species are shown to follow preferentially natural boundaries of their environment in a process called thigmotactism [9]. At the theoretical level, such boundary-mediated transport (sometimes called surface-mediated diffusion in the context of chemical physics) has raised a growing interest since the pioneering work of Adam and Delbrück, who recognized that diffusion of particles on low dimensional interfaces could speed up their search for interfacial targets [4]. More recent works in fact revealed that search times for interfacial targets for particles performing surfacemediated diffusion can be minimized by a proper tuning of their desorption rate from the surface [2,10–19], which could constitute a general mechanism of enhancement of reaction kinetics [12,13]. However, so far most of the modeling efforts have focused on the case of interfacial targets, i.e., targets that are located at an interface, as in the examples of target sequences on DNA [1–3,20] or catalytic sites on a functionalized surface [21]. In such cases, it is now well understood that surface-mediated diffusion, if fast enough, can effectively increase the reaction radius of the target and therefore accelerate the search. This argument, however, does not hold a priori for bulk targets, i.e., targets that are located in the interior of the domain. Indeed, during phases of diffusion at interfaces, bulk targets are not accessible. In particular, in the regime of low desorption rate from the surface, the search time for bulk targets diverges, in striking contrast with the case of interfacial targets. The benefit of 0031-9007=14=112(23)=230601(5)
phases at the interface is finally far from clear, and cannot be deduced from the analysis of interfacial targets. On the other hand, search processes for generic bulk targets in confined domains has raised a growing interest in the last few years even in the case of simple diffusion. In particular it was shown that geometric parameters, such as the target position, could play a key role in the search kinetics [22–25]. In this context, the case of bulk targets in interfacial systems does not seem to have been considered so far, despite its relevance in chemical physics and potentially animal behavior sciences. In this Letter, we introduce a minimal model of search process with a bulk target in an interfacial system. We consider a random searcher performing boundary-mediated diffusion in a 2D disk in the presence of a target of arbitrary position, and address the following question: Can boundary excursions help the searcher finding the target faster? We derive analytical expressions of the mean first-passage time (MFPT) of the searcher at the target, as well as of the global MFPT (GMFPT), defined as the MFPT averaged over the initial position of the searcher. These expressions are exact in the limit of small target size ϵ and accurate for a wide range of ϵ. This analysis reveals that both the MFPT and the GMFPT, which are key observables for quantifying the kinetics of the search process, can be minimized as a function of the mean duration of boundary excursions. From the theoretical point of view, these results provide an exact solution of the generalization to surface-mediated diffusion of the MFPT problem, which is a classical problem of random walk theory that has attracted a lot of attention in the last few years in the case of regular diffusion both in the mathematical [6,26–29] and physical literature [22–25]. It also constitutes an extension of previous studies of surface-mediated reaction kinetics, which all considered interfacial targets [2,11–13,15–18]. As we proceed to show, the case of bulk targets of arbitrary
230601-1
© 2014 American Physical Society
PRL 112, 230601 (2014)
week ending 13 JUNE 2014
PHYSICAL REVIEW LETTERS hTT iðrÞ ¼ 0;
r ∈ T;
ð3Þ
where the Laplace operators Δ1 ¼ ∂ 2θ =R2 and Δ2 ¼ ∂ 2r þ ∂ r =r þ ∂ 2θ =r2 act on the initial state variable r, equivalently denoted in this paper by its polar coordinates ðr; θÞ for convenience. To solve this mixed boundary value problem, we generalize the method developed in [25] in the case of standard Brownian diffusion to the case at hand of boundary-mediated diffusion. A first important technical step is to introduce the pseudo-Green function Hðr0 jrÞ defined by FIG. 1 (color online). Model of surface-mediated diffusion in a confining domain, with the target T (in red) in rT . The particle (in green), starting from rS , can reach T by diffusing only inside the bulk (in orange), or by using also surface diffusion phases (in blue).
Pst ðr0 Þ − δðr0 − rÞ ¼ D1 Δ1 Hðr0 jR; θÞ
position is in fact technically very different and requires a new approach. As for applications, these results put forward a generic mechanism of optimization of search kinetics for bulk targets by tuning the affinity of the searcher for interfaces, and as such could have important implications in chemical physics. In the context of behavioral ecology, it shows that following boundaries can accelerate a search process, and therefore suggests that animal thigmotactism could be kinetically beneficial [30]. As an archetype of confined interfacial systems, we consider a random searcher inside a disk D of radius R (Fig. 1), alternating phases of surface diffusion on the boundary ∂D with diffusion coefficient D1 and phases of bulk diffusion inside D with diffusion coefficient D2 . The time spent during each boundary phase is assumed to follow an exponential law with rate λ. Each time the searcher leaves the boundary, it is assumed to restart in the bulk at a distance a from the boundary (otherwise it would instantaneously rebind). Although formulated for any value of this parameter a ≤ R, in most situations of physical interest one has a ≪ R. We assume that the searcher instantaneously binds to the boundary upon encounter. We focus in this paper on the MFPT hTT iðrS Þ of a searcher starting from rS to∘ a spherical target T of radius ϵ, located in the interior D of D at an arbitrary position rT . In fact, as expected from [31,32], and checked numerically, the distribution of this FPT to a bulk target converges in the large domain size limit to a single exponential, in strong contrast to the case of interfacial targets (see [12]). The MFPT therefore fully characterizes the FPT statistics. Following earlier works [12,13], the MFPT can be shown to satisfy the following backward Fokker Planck equations [33,34]:
where Pst ðrÞ is the stationary probability that the searcher is at r, which can be shown to be given by ln x; r − 2 T D2 xT
ϵ 1 2 ln ð1 − xT Þ þ ; R 4
ð17Þ
ð15Þ
ð16Þ
where xT ≡ 1 − rT =R. Note that the right-hand side is always strictly positive for bulk targets, so that there is a nontrivial threshold value for ρD that depends on the target position and extension. The closer to the boundary the target, the lower this threshold value. In other words, for any target position (except perfectly centered targets), the GMFPT can be minimized provided that surface diffusion is fast enough, as quantified by Eq. (17). A lower bound hTT im of the GMFPT is obtained by taking first D1 → ∞ and λ → ∞, which leads in the large system size limit to hTT im ∝ R2 =D2 for targets located close to the boundary. Knowing that the GMFPT for bulk diffusion only is given in the large system size limit by hT¯T ib ∼ R2 =ð2D2 Þ lnðR=ϵÞ [25], one finds that boundary excursions can yield a significant gain for targets close to the boundary, as quantified in Fig. 3(b). Qualitatively, such minimization of the GMFPT can be understood using the analogy with intermittent processes [2,36]. Here bulk phases enable target detection, while surface phases do not, which seems to suggest that surface diffusion is not favorable for a rapid search. However, if surface diffusion is fast enough, it gives rapid access to remote areas, which is favorable. It has been shown on general grounds in the literature [2,36] that such intermittent trajectories can indeed minimize the MFPT to a target. Quantitatively, the optimal range of values of λ can be determined numerically by the analysis of the analytical expression of the GMFPT. We here provide scaling arguments that yield explicit expressions and underline the
230601-3
week ending 13 JUNE 2014
PHYSICAL REVIEW LETTERS
PRL 112, 230601 (2014) (a)
(b)
(c)
(d)
FIG. 2 (color online). MFPT hTT i from Eq. (16) for a ¼ 0.1 and D1 ¼ 1. (a) MFPT as a function of rs , with a centered target T, for D2 ¼ 1 and λ ¼ 10. (b) MFPT as a function of rT , with a quasicentered source in ðϵ; π=2Þ, for D2 ¼ 1 and ϵ ¼ 0.1. (c) MFPT as a function of λ, with the target T in ðR − a − ϵ; 0Þ and the source S in ðR − a − ϵ; 0.7πÞ, for ϵ ¼ 0.1. (d) MFPT as a function of λ, with the target T in ðR − a − ϵ; 0Þ and the source S in ðR − a − ϵ; θS Þ, for D2 ¼ 0.1 and ϵ ¼ 0.1.
robustness of the optimization for various domain shapes. Let us consider a generic 2D domain of area A and perimeter P. For sufficiently small target sizes ϵ, using the analogy with intermittent processes [2,36], the GMFPT can be written hTT i ≃ nðτ1 þ τ2 Þ, where n denotes the mean number of boundary and bulk excursions needed to find the target, τ1 ¼ 1=λ the mean duration of a boundary excursion, and τ2 ≃ aA=ðD2 PÞ the mean duration of a bulk excursion [37]. Using classical results on Brownian motion, the typical span of ffia bulk excursion is given by pffiffiffiffiffiffiffiffiffiffi a. In the regime a ≲ D1 =λ, successive bulk excursions, which in fact enable target detection, therefore typically do not overlap. Using again the analogy with intermittent processes, we deduce that n ∝ P=a. Altogether, this provides a simple approximation of the GMFPT in the regime λ ≲ D1 =a2 (in agreement with the exact expression derived above for circular domains) which can be written P 1 aA hTT i ∝ þ : a λ D2 P
ð18Þ
In the regime λ ≳ D1 =a2 , the GMFPT is expected to increase and reach its plateau value hTT ib . This analysis, in particular, predicts that the GMFPT is minimal, and in fact surprisingly enough independent of λ in the range D2 P=ðaAÞ ≲ λ ≲ D1 =a2 , which can in practice span several orders of magnitude, as confirmed in Fig. 3(c). Note
(a)
(b)
(c)
(d)
FIG. 3 (color online). (a) GMFPT hT¯T i as a function of λ and rT , for a ¼ 0.01, ϵ ¼ 0.1, D2 ¼ 0.1, and D1 ¼ 1. (b) Relative gain on the GMFPT (jhTT im − hTT ib j=hTT ib ) as a function of the target position rT , for a ¼ 0.01, D2 ¼ 0.1, and D1 ¼ 1. (c) GMFPT hTT i as a function of λ and ρD , for a target in rT ¼ R − a − ϵ, a ¼ ϵ ¼ 0.01, and D2 ¼ 0.1. (d) GMFPT hTT i as a function of λ, in the case of an ellipse (eccentricity 0.95, surface π), for D1 ¼ 104, D2 ¼ 0.1, and a ¼ ϵ ¼ 0.01. The case called A stands for the target located on the semimajor axis, at a distance a from the boundary, whereas the case called B stands for the target located on the semiminor axis, at a distance a from the boundary. The fit expression stands for the result of Eq. (18), with a fitted proportionality parameter.
that this feature is in striking contrast with the case of interfacial targets [12]. Interestingly, these arguments hold for general domain shapes, which reinforces the robustness of the results. Figure 3(d) confirms that, in the example of elliptic shapes, indeed (i) the GMFPT can be minimized, (ii) this minimum can be realized for a broad range of values of λ, and (iii) our scaling analysis provides a good approximate of the GMFPT up to a numerical factor. In conclusion, we have provided exact expressions of the MFPT and GMFPT to a bulk target for a random searcher that performs boundary-mediated diffusion in a circular domain. Although rather counterintuitive for bulk targets, it is found that boundary excursions, if fast enough, can accelerate the search process. A scaling formula extends these results to domains of arbitrary shapes and demonstrates their robustness. These results put forward a generic mechanism of optimization of search kinetics in interfacial systems, which could have important implications in chemical physics. It further shows that following boundaries can accelerate a search process, and therefore suggests that thigmotactic behaviors observed for various animal species could be kinetically beneficial. Support from the European Research Council starting Grant No. FPTOpt-277998 is acknowledged.
230601-4
PRL 112, 230601 (2014)
PHYSICAL REVIEW LETTERS
[1] O. G. Berg, R. B. Winter, and P. H. von Hippel, Biochemistry 20, 6929 (1981). [2] O. Benichou, C. Loverdo, M. Moreau, and R. Voituriez, Phys. Chem. Chem. Phys. 10, 7059 (2008). [3] M. Sheinman, O. Bénichou, Y. Kafri, and R. Voituriez, Rep. Prog. Phys. 75, 026601 (2012). [4] G. Adam and M. Delbrück, Reduction of Dimensionality in Biological Diffusion Processes (W.H. Freeman, San Francisco, 1968). [5] H. Sano and M. Tachiya, J. Chem. Phys. 75, 2870 (1981). [6] Z. Schuss, A. Singer, and D. Holcman, Proc. Natl. Acad. Sci. U.S.A. 104, 16098 (2007). [7] R. Walder, N. Nelson, and D. K. Schwartz, Phys. Rev. Lett. 107, 156102 (2011). [8] M. J. Skaug, J. Mabry, and D. K. Schwartz, Phys. Rev. Lett. 110, 256101 (2013). [9] C. Detrain, J. L. Deneubourg, and J. M. Pasteels, Information Processing in Social Insects (Birkhauser, Basel, 1999). [10] A. V. Chechkin, I. M. Zaid, M. A. Lomholt, I. M. Sokolov, and R. Metzler, Phys. Rev. E 79 (2009). [11] G. Oshanin, M. Tamm, and O. Vasilyev, J. Chem. Phys. 132, 235101 (2010). [12] O. Bénichou, D. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, Phys. Rev. Lett. 105, 150606 (2010). [13] O. Bénichou, D. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, J. Stat. Phys. 142, 657 (2011). [14] A. V. Chechkin, I. M. Zaid, M. A. Lomholt, I. M. Sokolov, and R. Metzler, J. Chem. Phys. 134, 204116 (2011). [15] F. Rojo and C. E. Budde, Phys. Rev. E 84 (2011). [16] F. Rojo, H. S. Wio, and C. E. Budde, Phys. Rev. E 86, 031105 (2012). [17] F. Rojo, C. E. Budde, Carlos, H. S. Wio, and C. E. Budde, Phys. Rev. E 87, 012115 (2013). [18] A. M. Berezhkovskii and A. V. Barzykin, J. Chem. Phys. 136, 054115 (2012). [19] J. F. Rupprecht, O. Bénichou, D. S. Grebenkov, and R. Voituriez, Phys. Rev. E 86, 041135 (2012). [20] O. Bénichou, C. Chevalier, B. Meyer, and R. Voituriez, Phys. Rev. Lett. 106, 038102 (2011).
week ending 13 JUNE 2014
[21] G. C. Bond, Heterogeneous Catalysis: Principles and Applications (Clarendon, Oxford, 1987). [22] S. Condamin, O. Benichou, and M. Moreau, Phys. Rev. Lett. 95, 260601 (2005). [23] S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, Nature (London) 450, 77 (2007). [24] S. Condamin, V. Tejedor, R. Voituriez, O. Benichou, and J. Klafter, Proc. Natl. Acad. Sci. U.S.A. 105, 5675 (2008). [25] C. Chevalier, O. Bénichou, B. Meyer, and R. Voituriez, J. Phys. A 44, 025002 (2011). [26] M. J. Ward and J. B. Keller, SIAM J. Appl. Math. 53, 770 (1993). [27] S. Pillay, M. J. Ward, A. Peirce, and T. Kolokolnikov, Multiscale Model. Simul. 8, 803 (2010). [28] A. F. Cheviakov, M. J. Ward, and R. Straube, Multiscale Model. Simul. 8, 836 (2010). [29] A. Singer, Z. Schuss, and D. Holcman, J. Stat. Phys. 122, 465 (2006). [30] Here we neglect collective effects, described, for example, in [38,39]. [31] O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Nat. Chem. 2, 472 (2010). [32] B. Meyer, C. Chevalier, R. Voituriez, and O. Bénichou, Phys. Rev. E 83, 051116 (2011). [33] S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, England, 2001). [34] H.Risken, The Fokker-Planck Equation: Method of Solution and Applications (Springer-Verlag, Berlin, 1996). [35] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.230601 for explicit expressions of pseudo-Green functions and details of derivation. [36] O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Rev. Mod. Phys. 83, 81 (2011). [37] O. Bénichou, M. Coppey, M. Moreau, P. H. Suet, and R. Voituriez, Europhys. Lett. 70, 42 (2005). [38] K. Bhattacharya and T. Vicsek, arXiv:1311.2169. [39] R. Martínez-García, J. M. Calabrese, T. Mueller, K. A. Olson, and C. López, Phys. Rev. Lett. 110, 248106 (2013).
230601-5
PHYSICAL REVIEW LETTERS
PRL 112, 230601 (2014)
Accelerating search kinetics by following boundaries 1
T. Calandre,1 O. Bénichou,1 and R. Voituriez1,2 Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), CNRS / UPMC, 4 Place Jussieu, 75255 Paris Cedex 2 Laboratoire Jean Perrin (FRE 3231) CNRS /UPMC, 4 Place Jussieu, 75255 Paris Cedex (Received 4 March 2014; published 9 June 2014) We derive exact expressions of the mean first-passage time to a bulk target for a random searcher that performs boundary-mediated diffusion in a circular domain. Although nonintuitive for bulk targets, it is found that boundary excursions, if fast enough, can minimize the search time. A scaling analysis generalizes these findings to domains of arbitrary shapes and underlines their robustness. Overall, these results provide a generic mechanism of optimization of search kinetics in interfacial systems, which could have important implications in chemical physics. In the context of animal behavior sciences, it shows that following the boundaries of a domain can accelerate a search process, and therefore suggests that thigmotactism could be a kinetically efficient behavior. DOI: 10.1103/PhysRevLett.112.230601
PACS numbers: 05.40.Fb, 82.20.−w, 87.23.−n
In interfacial systems, such as porous media, or biopolymer [1–3], micellar and colloidal [4–6] systems, the transport of a generic tracer particle combines diffusion phases in the bulk and diffusion phases at interfaces due to transient binding events, which can now be observed at the single molecule scale [7,8]. Interestingly, similar intermittent transport patterns have also been observed at a larger scale in the context of animal movement, where numerous species are shown to follow preferentially natural boundaries of their environment in a process called thigmotactism [9]. At the theoretical level, such boundary-mediated transport (sometimes called surface-mediated diffusion in the context of chemical physics) has raised a growing interest since the pioneering work of Adam and Delbrück, who recognized that diffusion of particles on low dimensional interfaces could speed up their search for interfacial targets [4]. More recent works in fact revealed that search times for interfacial targets for particles performing surfacemediated diffusion can be minimized by a proper tuning of their desorption rate from the surface [2,10–19], which could constitute a general mechanism of enhancement of reaction kinetics [12,13]. However, so far most of the modeling efforts have focused on the case of interfacial targets, i.e., targets that are located at an interface, as in the examples of target sequences on DNA [1–3,20] or catalytic sites on a functionalized surface [21]. In such cases, it is now well understood that surface-mediated diffusion, if fast enough, can effectively increase the reaction radius of the target and therefore accelerate the search. This argument, however, does not hold a priori for bulk targets, i.e., targets that are located in the interior of the domain. Indeed, during phases of diffusion at interfaces, bulk targets are not accessible. In particular, in the regime of low desorption rate from the surface, the search time for bulk targets diverges, in striking contrast with the case of interfacial targets. The benefit of 0031-9007=14=112(23)=230601(5)
phases at the interface is finally far from clear, and cannot be deduced from the analysis of interfacial targets. On the other hand, search processes for generic bulk targets in confined domains has raised a growing interest in the last few years even in the case of simple diffusion. In particular it was shown that geometric parameters, such as the target position, could play a key role in the search kinetics [22–25]. In this context, the case of bulk targets in interfacial systems does not seem to have been considered so far, despite its relevance in chemical physics and potentially animal behavior sciences. In this Letter, we introduce a minimal model of search process with a bulk target in an interfacial system. We consider a random searcher performing boundary-mediated diffusion in a 2D disk in the presence of a target of arbitrary position, and address the following question: Can boundary excursions help the searcher finding the target faster? We derive analytical expressions of the mean first-passage time (MFPT) of the searcher at the target, as well as of the global MFPT (GMFPT), defined as the MFPT averaged over the initial position of the searcher. These expressions are exact in the limit of small target size ϵ and accurate for a wide range of ϵ. This analysis reveals that both the MFPT and the GMFPT, which are key observables for quantifying the kinetics of the search process, can be minimized as a function of the mean duration of boundary excursions. From the theoretical point of view, these results provide an exact solution of the generalization to surface-mediated diffusion of the MFPT problem, which is a classical problem of random walk theory that has attracted a lot of attention in the last few years in the case of regular diffusion both in the mathematical [6,26–29] and physical literature [22–25]. It also constitutes an extension of previous studies of surface-mediated reaction kinetics, which all considered interfacial targets [2,11–13,15–18]. As we proceed to show, the case of bulk targets of arbitrary
230601-1
© 2014 American Physical Society
PRL 112, 230601 (2014)
week ending 13 JUNE 2014
PHYSICAL REVIEW LETTERS hTT iðrÞ ¼ 0;
r ∈ T;
ð3Þ
where the Laplace operators Δ1 ¼ ∂ 2θ =R2 and Δ2 ¼ ∂ 2r þ ∂ r =r þ ∂ 2θ =r2 act on the initial state variable r, equivalently denoted in this paper by its polar coordinates ðr; θÞ for convenience. To solve this mixed boundary value problem, we generalize the method developed in [25] in the case of standard Brownian diffusion to the case at hand of boundary-mediated diffusion. A first important technical step is to introduce the pseudo-Green function Hðr0 jrÞ defined by FIG. 1 (color online). Model of surface-mediated diffusion in a confining domain, with the target T (in red) in rT . The particle (in green), starting from rS , can reach T by diffusing only inside the bulk (in orange), or by using also surface diffusion phases (in blue).
Pst ðr0 Þ − δðr0 − rÞ ¼ D1 Δ1 Hðr0 jR; θÞ
position is in fact technically very different and requires a new approach. As for applications, these results put forward a generic mechanism of optimization of search kinetics for bulk targets by tuning the affinity of the searcher for interfaces, and as such could have important implications in chemical physics. In the context of behavioral ecology, it shows that following boundaries can accelerate a search process, and therefore suggests that animal thigmotactism could be kinetically beneficial [30]. As an archetype of confined interfacial systems, we consider a random searcher inside a disk D of radius R (Fig. 1), alternating phases of surface diffusion on the boundary ∂D with diffusion coefficient D1 and phases of bulk diffusion inside D with diffusion coefficient D2 . The time spent during each boundary phase is assumed to follow an exponential law with rate λ. Each time the searcher leaves the boundary, it is assumed to restart in the bulk at a distance a from the boundary (otherwise it would instantaneously rebind). Although formulated for any value of this parameter a ≤ R, in most situations of physical interest one has a ≪ R. We assume that the searcher instantaneously binds to the boundary upon encounter. We focus in this paper on the MFPT hTT iðrS Þ of a searcher starting from rS to∘ a spherical target T of radius ϵ, located in the interior D of D at an arbitrary position rT . In fact, as expected from [31,32], and checked numerically, the distribution of this FPT to a bulk target converges in the large domain size limit to a single exponential, in strong contrast to the case of interfacial targets (see [12]). The MFPT therefore fully characterizes the FPT statistics. Following earlier works [12,13], the MFPT can be shown to satisfy the following backward Fokker Planck equations [33,34]:
where Pst ðrÞ is the stationary probability that the searcher is at r, which can be shown to be given by ln x; r − 2 T D2 xT
ϵ 1 2 ln ð1 − xT Þ þ ; R 4
ð17Þ
ð15Þ
ð16Þ
where xT ≡ 1 − rT =R. Note that the right-hand side is always strictly positive for bulk targets, so that there is a nontrivial threshold value for ρD that depends on the target position and extension. The closer to the boundary the target, the lower this threshold value. In other words, for any target position (except perfectly centered targets), the GMFPT can be minimized provided that surface diffusion is fast enough, as quantified by Eq. (17). A lower bound hTT im of the GMFPT is obtained by taking first D1 → ∞ and λ → ∞, which leads in the large system size limit to hTT im ∝ R2 =D2 for targets located close to the boundary. Knowing that the GMFPT for bulk diffusion only is given in the large system size limit by hT¯T ib ∼ R2 =ð2D2 Þ lnðR=ϵÞ [25], one finds that boundary excursions can yield a significant gain for targets close to the boundary, as quantified in Fig. 3(b). Qualitatively, such minimization of the GMFPT can be understood using the analogy with intermittent processes [2,36]. Here bulk phases enable target detection, while surface phases do not, which seems to suggest that surface diffusion is not favorable for a rapid search. However, if surface diffusion is fast enough, it gives rapid access to remote areas, which is favorable. It has been shown on general grounds in the literature [2,36] that such intermittent trajectories can indeed minimize the MFPT to a target. Quantitatively, the optimal range of values of λ can be determined numerically by the analysis of the analytical expression of the GMFPT. We here provide scaling arguments that yield explicit expressions and underline the
230601-3
week ending 13 JUNE 2014
PHYSICAL REVIEW LETTERS
PRL 112, 230601 (2014) (a)
(b)
(c)
(d)
FIG. 2 (color online). MFPT hTT i from Eq. (16) for a ¼ 0.1 and D1 ¼ 1. (a) MFPT as a function of rs , with a centered target T, for D2 ¼ 1 and λ ¼ 10. (b) MFPT as a function of rT , with a quasicentered source in ðϵ; π=2Þ, for D2 ¼ 1 and ϵ ¼ 0.1. (c) MFPT as a function of λ, with the target T in ðR − a − ϵ; 0Þ and the source S in ðR − a − ϵ; 0.7πÞ, for ϵ ¼ 0.1. (d) MFPT as a function of λ, with the target T in ðR − a − ϵ; 0Þ and the source S in ðR − a − ϵ; θS Þ, for D2 ¼ 0.1 and ϵ ¼ 0.1.
robustness of the optimization for various domain shapes. Let us consider a generic 2D domain of area A and perimeter P. For sufficiently small target sizes ϵ, using the analogy with intermittent processes [2,36], the GMFPT can be written hTT i ≃ nðτ1 þ τ2 Þ, where n denotes the mean number of boundary and bulk excursions needed to find the target, τ1 ¼ 1=λ the mean duration of a boundary excursion, and τ2 ≃ aA=ðD2 PÞ the mean duration of a bulk excursion [37]. Using classical results on Brownian motion, the typical span of ffia bulk excursion is given by pffiffiffiffiffiffiffiffiffiffi a. In the regime a ≲ D1 =λ, successive bulk excursions, which in fact enable target detection, therefore typically do not overlap. Using again the analogy with intermittent processes, we deduce that n ∝ P=a. Altogether, this provides a simple approximation of the GMFPT in the regime λ ≲ D1 =a2 (in agreement with the exact expression derived above for circular domains) which can be written P 1 aA hTT i ∝ þ : a λ D2 P
ð18Þ
In the regime λ ≳ D1 =a2 , the GMFPT is expected to increase and reach its plateau value hTT ib . This analysis, in particular, predicts that the GMFPT is minimal, and in fact surprisingly enough independent of λ in the range D2 P=ðaAÞ ≲ λ ≲ D1 =a2 , which can in practice span several orders of magnitude, as confirmed in Fig. 3(c). Note
(a)
(b)
(c)
(d)
FIG. 3 (color online). (a) GMFPT hT¯T i as a function of λ and rT , for a ¼ 0.01, ϵ ¼ 0.1, D2 ¼ 0.1, and D1 ¼ 1. (b) Relative gain on the GMFPT (jhTT im − hTT ib j=hTT ib ) as a function of the target position rT , for a ¼ 0.01, D2 ¼ 0.1, and D1 ¼ 1. (c) GMFPT hTT i as a function of λ and ρD , for a target in rT ¼ R − a − ϵ, a ¼ ϵ ¼ 0.01, and D2 ¼ 0.1. (d) GMFPT hTT i as a function of λ, in the case of an ellipse (eccentricity 0.95, surface π), for D1 ¼ 104, D2 ¼ 0.1, and a ¼ ϵ ¼ 0.01. The case called A stands for the target located on the semimajor axis, at a distance a from the boundary, whereas the case called B stands for the target located on the semiminor axis, at a distance a from the boundary. The fit expression stands for the result of Eq. (18), with a fitted proportionality parameter.
that this feature is in striking contrast with the case of interfacial targets [12]. Interestingly, these arguments hold for general domain shapes, which reinforces the robustness of the results. Figure 3(d) confirms that, in the example of elliptic shapes, indeed (i) the GMFPT can be minimized, (ii) this minimum can be realized for a broad range of values of λ, and (iii) our scaling analysis provides a good approximate of the GMFPT up to a numerical factor. In conclusion, we have provided exact expressions of the MFPT and GMFPT to a bulk target for a random searcher that performs boundary-mediated diffusion in a circular domain. Although rather counterintuitive for bulk targets, it is found that boundary excursions, if fast enough, can accelerate the search process. A scaling formula extends these results to domains of arbitrary shapes and demonstrates their robustness. These results put forward a generic mechanism of optimization of search kinetics in interfacial systems, which could have important implications in chemical physics. It further shows that following boundaries can accelerate a search process, and therefore suggests that thigmotactic behaviors observed for various animal species could be kinetically beneficial. Support from the European Research Council starting Grant No. FPTOpt-277998 is acknowledged.
230601-4
PRL 112, 230601 (2014)
PHYSICAL REVIEW LETTERS
[1] O. G. Berg, R. B. Winter, and P. H. von Hippel, Biochemistry 20, 6929 (1981). [2] O. Benichou, C. Loverdo, M. Moreau, and R. Voituriez, Phys. Chem. Chem. Phys. 10, 7059 (2008). [3] M. Sheinman, O. Bénichou, Y. Kafri, and R. Voituriez, Rep. Prog. Phys. 75, 026601 (2012). [4] G. Adam and M. Delbrück, Reduction of Dimensionality in Biological Diffusion Processes (W.H. Freeman, San Francisco, 1968). [5] H. Sano and M. Tachiya, J. Chem. Phys. 75, 2870 (1981). [6] Z. Schuss, A. Singer, and D. Holcman, Proc. Natl. Acad. Sci. U.S.A. 104, 16098 (2007). [7] R. Walder, N. Nelson, and D. K. Schwartz, Phys. Rev. Lett. 107, 156102 (2011). [8] M. J. Skaug, J. Mabry, and D. K. Schwartz, Phys. Rev. Lett. 110, 256101 (2013). [9] C. Detrain, J. L. Deneubourg, and J. M. Pasteels, Information Processing in Social Insects (Birkhauser, Basel, 1999). [10] A. V. Chechkin, I. M. Zaid, M. A. Lomholt, I. M. Sokolov, and R. Metzler, Phys. Rev. E 79 (2009). [11] G. Oshanin, M. Tamm, and O. Vasilyev, J. Chem. Phys. 132, 235101 (2010). [12] O. Bénichou, D. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, Phys. Rev. Lett. 105, 150606 (2010). [13] O. Bénichou, D. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, J. Stat. Phys. 142, 657 (2011). [14] A. V. Chechkin, I. M. Zaid, M. A. Lomholt, I. M. Sokolov, and R. Metzler, J. Chem. Phys. 134, 204116 (2011). [15] F. Rojo and C. E. Budde, Phys. Rev. E 84 (2011). [16] F. Rojo, H. S. Wio, and C. E. Budde, Phys. Rev. E 86, 031105 (2012). [17] F. Rojo, C. E. Budde, Carlos, H. S. Wio, and C. E. Budde, Phys. Rev. E 87, 012115 (2013). [18] A. M. Berezhkovskii and A. V. Barzykin, J. Chem. Phys. 136, 054115 (2012). [19] J. F. Rupprecht, O. Bénichou, D. S. Grebenkov, and R. Voituriez, Phys. Rev. E 86, 041135 (2012). [20] O. Bénichou, C. Chevalier, B. Meyer, and R. Voituriez, Phys. Rev. Lett. 106, 038102 (2011).
week ending 13 JUNE 2014
[21] G. C. Bond, Heterogeneous Catalysis: Principles and Applications (Clarendon, Oxford, 1987). [22] S. Condamin, O. Benichou, and M. Moreau, Phys. Rev. Lett. 95, 260601 (2005). [23] S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, Nature (London) 450, 77 (2007). [24] S. Condamin, V. Tejedor, R. Voituriez, O. Benichou, and J. Klafter, Proc. Natl. Acad. Sci. U.S.A. 105, 5675 (2008). [25] C. Chevalier, O. Bénichou, B. Meyer, and R. Voituriez, J. Phys. A 44, 025002 (2011). [26] M. J. Ward and J. B. Keller, SIAM J. Appl. Math. 53, 770 (1993). [27] S. Pillay, M. J. Ward, A. Peirce, and T. Kolokolnikov, Multiscale Model. Simul. 8, 803 (2010). [28] A. F. Cheviakov, M. J. Ward, and R. Straube, Multiscale Model. Simul. 8, 836 (2010). [29] A. Singer, Z. Schuss, and D. Holcman, J. Stat. Phys. 122, 465 (2006). [30] Here we neglect collective effects, described, for example, in [38,39]. [31] O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Nat. Chem. 2, 472 (2010). [32] B. Meyer, C. Chevalier, R. Voituriez, and O. Bénichou, Phys. Rev. E 83, 051116 (2011). [33] S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, England, 2001). [34] H.Risken, The Fokker-Planck Equation: Method of Solution and Applications (Springer-Verlag, Berlin, 1996). [35] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.230601 for explicit expressions of pseudo-Green functions and details of derivation. [36] O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Rev. Mod. Phys. 83, 81 (2011). [37] O. Bénichou, M. Coppey, M. Moreau, P. H. Suet, and R. Voituriez, Europhys. Lett. 70, 42 (2005). [38] K. Bhattacharya and T. Vicsek, arXiv:1311.2169. [39] R. Martínez-García, J. M. Calabrese, T. Mueller, K. A. Olson, and C. López, Phys. Rev. Lett. 110, 248106 (2013).
230601-5
